{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Maximum Likelihood Estimate versus Maximum-a-posteriori Estimate\n",
    "\n",
    "## A simple illustrative example with coin toss.\n",
    "\n",
    "Let’s say that we have a coin and each time we toss the coin we can get heads (=H) or Tails (=T).\n",
    "In one example we toss the coin twice and get two heads and one tail in a row.  Now our data, $D=[H,H, T]$.\n",
    "Let θ be the parameter we want to estimate, where θ is the probability of getting heads, $P(H)=θ$.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Case 1: ML estimate\n",
    "\n",
    "\n",
    "$θ_{ML}=\\arg⁡ \\underset{θ}{\\max}⁡ P(data|θ)$\n",
    "\n",
    "$P(data│θ)=P(D|θ)=P(D;θ)=P(D=[H,H,T];θ)=(θ)(θ)(1-θ)=θ^2-θ^3$\n",
    "\n",
    "The $θ$ that maximises $θ^2-θ^3$, is $θ_{MLE}=\\tfrac{2}{3}$. \n",
    "But this is over fitting in a sense, because it is assuming, based on our limited data, that we will get Heads more often than Tails (The Maximum Likelihood estimate says that the θ that maximises the probability we got this exact example data set, $D$, is  $θ=\\tfrac{2}{3}$, and thus we should expect to get heads $\\tfrac{2}{3}$ of the time in future coin tosses).  $P(D│θ)$ is a likelihood since it is based on limited data and $\\int_0^1 P(D|θ) dθ\\ne1$.  We only need to integrate between 0 and 1 since we know $θ\\in(0,1)$. \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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7cyGHsXM2UM3DjQ9Hda7USUWzXevg6tzTrSlz1x5k57G/XM9qVsxILOuBViLS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\noxGrdxRScuSoK9uzwhImRyuV0Z9s4O7Xl9E0qR4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      "text/plain": [
       "<matplotlib.figure.Figure at 0x4ad44a8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$$P(H)=\\tfrac{2}{3} \\text{  (The probability of getting heads in our coin toss is} \\tfrac{2}{3} \\text{)}$$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "import pandas as pd\n",
    "import matplotlib.pyplot as plt\n",
    "from IPython.display import display, Math, Latex\n",
    "\n",
    "\n",
    "theta = np.arange(0.0, 1.0, 0.001)\n",
    "p_likelihood = (theta**2-theta**3)\n",
    "plt.plot(theta, p_likelihood)\n",
    "plt.plot([2/3,2/3], [0,max(p_likelihood)])\n",
    "plt.axis([0, 1, 0, 0.15])\n",
    "plt.grid(True)\n",
    "plt.annotate('Maximum Likelihood Estimate', xy=(2/3, 0), xytext=(2/3-0.3, 0.03),\n",
    "            arrowprops=dict(facecolor='darkorange', shrink=2),\n",
    "            )\n",
    "plt.annotate(r'$\\hat{\\theta}_{MLE}$', (0,0), (220, -3), xycoords='axes fraction', textcoords='offset points', va='top')\n",
    "plt.annotate('Maximum', xy=(2/3, max(p_likelihood)), xytext=(0.35, 0.9*max(p_likelihood)),\n",
    "            arrowprops=dict(facecolor='b', shrink=2),\n",
    "            )\n",
    "plt.xlabel(r'$\\theta$')\n",
    "plt.ylabel(r'$P(D│θ)$')\n",
    "plt.title('Likelihood Function for {H, H, T} Coin Toss sequence')\n",
    "plt.show()\n",
    "\n",
    "display(Math(r'P(H)=\\tfrac{2}{3} \\text{  (The probability of getting heads in our coin toss is} \\tfrac{2}{3} \\text{)}'))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Case 2: Bayesian MAP estimate\n",
    "\n",
    "Bayesian estimation assumes that we have prior knowledge about the thing we are trying to estimate, and Bayesian estimation assumes that our estimate is random rather than deterministic. Let’s use our prior knowledge that we expect a fair coin, that is, $θ=\\tfrac{1}{2}$. But this is not random, so instead, let’s make θ random by taking a leap of imagination and supposing that we have a box of coins and that we conduct each coin toss with different coin where some of our coins are a bit dodgy.  Maybe of coins are unusually weighted, or some have become a bit warped with age, with the result that a Gaussian prior is reasonable for our data.  That is, a Gaussian probability density function with mean=0.5.  (i.e the expected value of θ, $E[θ]=\\tfrac{1}{2}$, and our prior knowledge of coin tossing is a Gaussian pdf),\n",
    "\n",
    "$P(θ)=\\tfrac{1}{\\sqrt{2π\\sigma^2}} e^{\\tfrac{-(θ-0.5)^2}{2\\sigma^2}}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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MNh240ruKqjuQraqbMCZSZs2CpCTo1y/oSCKjVSs3trklDeMjv6qnGgGvee0a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\nMcaEzZKGMcaYsFnSMMYYE7b/D1kQzgcR5A1KAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x9dfd390>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "#plot gaussian prior\n",
    "\n",
    "sig = 0.1\n",
    "p_theta = 1/np.sqrt(2*np.pi*sig**2)*np.exp((-(theta-0.5)**2)/(2*sig**2))\n",
    "plt.plot(theta, p_theta, 'red')\n",
    "plt.plot([0.5,0.5], [0,max(p_theta)], 'darkorange')\n",
    "plt.axis([0, 1, 0, 4.1])\n",
    "plt.grid(True)\n",
    "plt.annotate(r'$E[\\theta]$', xy=(0.5, 0), xytext=(0.5-0.15, 0.5),\n",
    "            arrowprops=dict(facecolor='darkorange', shrink=2),\n",
    "            )\n",
    "plt.annotate('Maximum', xy=(0.5, 4), xytext=(0.23, 3.8),\n",
    "            arrowprops=dict(facecolor='red', shrink=2),\n",
    "            )\n",
    "plt.xlabel(r'$\\theta$')\n",
    "plt.ylabel(r'$P(θ)$')\n",
    "plt.title('Gaussian Prior Propability of getting Heads in Coin Toss')\n",
    "plt.show()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The maximum a-posteriori estimate is, $θ_{MAP}=\\arg \\max_θ⁡ P(θ|D)$.\n",
    "\n",
    "Using Bayes Rule, $P(θ│D)$ is proportional to the product of the likelihood from our measured data and the prior probability. (Color coding of the probabilty and likelihood distributions corresponds to their associated plots). \n",
    "\n",
    "$\\Large {\\color{purple}{P(θ│D)}}=\\frac{   {\\color{blue}{P(θ|D)}}    {\\color{red}{P(θ)}}  }{  {\\color{green}{\\int P(θ|D)P(θ) dθ}}} =  \n",
    "\\frac{   {\\color{blue}{\\left(θ^2-θ^3\\right)}}    {\\color{red}{\\left(\\tfrac{1}{\\sqrt{2π\\sigma^2}} e^{\\tfrac{-(θ-0.5)^2}{2\\sigma^2}}\\right)}}  }{  {\\color{green}{\\left(\\text{constant normalising term}\\right)}}}$\n",
    "\n",
    "Here, $P(θ|D)$ is a probability since  $\\int_0^1 P(θ|D) dθ=1$ (the normalising term makes sure of this).  The denominator $\\int P(θ|D)P(θ) dθ$ evaluates to a constant, therefore we can neglect this from the maximisation mathematics and $\\arg \\max_θ P(θ│D)$ is simply equal to $\\arg \\max_θ P(θ|D)P(θ)$.\n",
    "\n",
    "The plot below illustrates $P(θ│D)$.  The plot shows that $θ_{MAP}= 0.518$.  The prior probability $P(θ)$ tells us that the expected value of $θ$ is $\\tfrac{1}{2}$, $\\left(E[θ]=\\tfrac{1}{2}\\right)$, with some margin of error due to our dodgy coins, where this error margin is mathematically modelled by the variance in our Gaussian prior.  Our measurment data indicates that heads were twice as likely as tails $\\left(E[θ]=\\tfrac{2}{3}\\right)$.  The MAP estimate makes the best decision (statistically best decision) between the prior knowledge and the data we measured.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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PkbJLewVU1UMTh1K2QxsOkXk8s9a2bxQIHREKQPyPWl2lqocmDqVsBT+0BT+8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\nC4dzzrmMeOFwzjmXkf8DsJaVSzSscLwAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x9df66d8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$$\\text{The probability of Heads, } P(H) \\text{using the Bayesian Maximum A-Posteriori Estimate is } 0.518$$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# plot product of gaussian prior and likelihood and find the Bayesian Maximum A-Posteriori Estimate \n",
    "p_posterior = p_theta*p_likelihood\n",
    "plt.plot(theta, p_posterior, 'purple')\n",
    "max_p = max(p_posterior)\n",
    "max_ind = np.argmax(p_posterior)\n",
    "\n",
    "plt.plot([theta[max_ind],theta[max_ind]], [0,max_p], 'darkorange')\n",
    "plt.axis([0, 1, 0, 0.53])\n",
    "plt.grid(True)\n",
    "plt.annotate('Maximum A-Posteriori Estimate', xy=(theta[max_ind], 0), xytext=(theta[max_ind]-0.15, 0.1),\n",
    "            arrowprops=dict(facecolor='darkorange', shrink=2),\n",
    "            )\n",
    "plt.annotate('Maximum', xy=(theta[max_ind], max_p), xytext=(0.23, max_p-0.07),\n",
    "            arrowprops=dict(facecolor='purple', shrink=2),\n",
    "            )\n",
    "plt.annotate(r'$\\hat{\\theta}_{MAP}$', (0,0), (170, -3), xycoords='axes fraction', textcoords='offset points', va='top')\n",
    "plt.xlabel(r'$\\theta$')\n",
    "plt.ylabel(r'$P(θ|D)$')\n",
    "plt.title('Posterior Propability of getting Heads in Coin Toss')\n",
    "plt.show()\n",
    "\n",
    "\n",
    "#print (\"The probability of Heads, P(H) using the Bayesian Maximum A-Posteriori Estimate is %s\" % theta[max_ind])\n",
    "display(Math(r'\\text{The probability of Heads, } P(H) \\text{using the Bayesian Maximum A-Posteriori Estimate is } %s' % theta[max_ind]))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Interactive illustrative estimator with coin toss using widgets"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Choose a sequence of heads and tails using test field entry box, and the variance of the prior using the slider below to view and compare the Maximum Likelihood Estimate and the Bayesian Maximum A-Posteriori estimate:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$1. \\text{ Enter data sequence of coin coss outcomes (e.g. HHT for heads heads tails) @userinput}$$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$$2. \\text{ Choose } \\sigma \\text{ of the gaussian prior propability of getting Heads in coin toss where } \\sigma^2\\text{=variance}$$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "You entered: HHT\n",
      "and sigma = 0.19\n"
     ]
    },
    {
     "data": {
      "image/png": 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Mw/AyS55fwvZ527nx9Rup36u+2+V+//13qlL1iukxyTFc2HqBewffi1KKkpVL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\nyRvIqpTappQ6qJTqYbHoki8p7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      "text/plain": [
       "<matplotlib.figure.Figure at 0xb456ef0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$$\\text{The probability of Heads, } P(H)=\\theta,\\text{ using the Maximum Likelihood Estimate is } 0.667$$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$$\\text{The expected value of } P(H)=\\theta \\text{ using the Prior Probability is } 0.5$$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$$\\text{The probability of Heads, }  P(H)=\\theta, \\text{ using the Bayesian Maximum A-Posteriori Estimate is } 0.551$$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from ipywidgets import * # * => import everything\n",
    "import ipywidgets as widgets\n",
    "\n",
    "#user enter Heads or Tails as text sequence of H and T\n",
    "#user enters sigma value for prior distribution\n",
    "\n",
    "def plt_ML_BAY_EST(userinput, sigma):\n",
    "    display(Math(r'1. \\text{ Enter data sequence of coin coss outcomes (e.g. HHT for heads heads tails) @userinput}'))\n",
    "    display(Math(r'2. \\text{ Choose } \\sigma \\text{ of the gaussian prior propability of getting Heads in coin toss where } \\sigma^2\\text{=variance}'))\n",
    "    plt.figure(figsize=(16, 4))\n",
    "    sig=sigma\n",
    "    theta = np.arange(0.0, 1.0, 0.001)\n",
    "    print('You entered:', userinput)\n",
    "    print('and sigma =', sig)\n",
    "    UsrInptList = list(userinput)\n",
    "    UsrInptArray = np.array(UsrInptList)\n",
    "    p_likelihood = np.ones([max(theta.shape),UsrInptArray.shape[0]]) #preallocate\n",
    "\n",
    "    \n",
    "    ##FIRST PLOT ################################################\n",
    "    for i in range(UsrInptArray.shape[0]):\n",
    "        outcome=UsrInptArray[i]\n",
    "        if outcome == 'H': \n",
    "            p_likelihood[:,i]=theta\n",
    "        elif outcome == 'T': \n",
    "            p_likelihood[:,i]=1-theta\n",
    "\n",
    "    p_Likelihood = p_likelihood.prod(axis=1)\n",
    "    maxL = max(p_Likelihood)\n",
    "    maxL_ind = np.argmax(p_Likelihood)\n",
    "    plt.subplot(131)\n",
    "    plt.plot(theta, p_Likelihood)\n",
    "    plt.plot([theta[maxL_ind],theta[maxL_ind]], [0,maxL])\n",
    "    plt.axis([0, 1, 0, maxL+0.03*maxL])\n",
    "    plt.grid(True)\n",
    "    plt.annotate(r'Maximum Likelihood Estimate, $\\hat{\\theta}_{MLE}$', xy=(theta[maxL_ind], 0), \n",
    "            xytext=(theta[maxL_ind]-0.3*theta[maxL_ind], maxL*0.25),\n",
    "            arrowprops=dict(facecolor='darkorange', shrink=2),\n",
    "            )\n",
    "    plt.annotate('Maximum', xy=(theta[maxL_ind], max(p_Likelihood)), xytext=(0.35, 0.9*max(p_Likelihood)),\n",
    "            arrowprops=dict(facecolor='b', shrink=2),\n",
    "            )\n",
    "    plt.xlabel(r'$\\theta$')\n",
    "    plt.ylabel(r'$P(D│θ)$')\n",
    "    plt.title('Likelihood Function')\n",
    " \n",
    "    ##SECOND PLOT ################################################\n",
    "    plt.subplot(132)\n",
    "    p_theta = 1/np.sqrt(2*np.pi*sig**2)*np.exp((-(theta-0.5)**2)/(2*sig**2))\n",
    "    plt.plot(theta, p_theta, 'red')\n",
    "    plt.plot([0.5,0.5], [0,max(p_theta)], 'darkorange')\n",
    "    plt.axis([0, 1, 0, max(p_theta)+0.03*max(p_theta)])\n",
    "    plt.grid(True)\n",
    "    plt.annotate(r'$E[\\theta]$', xy=(0.5, 0), xytext=(0.5-0.15, 0.25*max(p_theta)),\n",
    "                arrowprops=dict(facecolor='darkorange', shrink=2),\n",
    "                )\n",
    "    plt.annotate('Maximum', xy=(0.5, max(p_theta)), xytext=(0.2, max(p_theta)-0.1*max(p_theta)),\n",
    "                arrowprops=dict(facecolor='red', shrink=2),\n",
    "                )\n",
    "    plt.xlabel(r'$\\theta$')\n",
    "    plt.ylabel(r'$P(θ)$')\n",
    "    plt.title('Gaussian Prior Probability of getting Heads')\n",
    "    \n",
    "    \n",
    "    ##THIRD PLOT ################################################\n",
    "    plt.subplot(133)\n",
    "    p_posterior = p_theta*p_Likelihood\n",
    "    plt.plot(theta, p_posterior, 'purple')\n",
    "    max_p3 = max(p_posterior)\n",
    "    max_ind3 = np.argmax(p_posterior)\n",
    "\n",
    "    plt.plot([theta[max_ind3],theta[max_ind3]], [0,max_p3], 'darkorange')\n",
    "    plt.axis([0, 1, 0, max_p3+0.03*max_p3])\n",
    "    plt.grid(True)\n",
    "    plt.annotate(r'Maximum A-Posteriori Estimate, $\\hat{\\theta}_{MAP}$', xy=(theta[max_ind3], 0), \n",
    "                 xytext=(theta[max_ind3]+0.3-((1-theta[max_ind3])/theta[max_ind3]), max_p3*0.2),\n",
    "                arrowprops=dict(facecolor='darkorange', shrink=2),\n",
    "                ) \n",
    "    plt.annotate('Maximum', xy=(theta[max_ind3], max_p3), xytext=(theta[max_ind3]-0.3*theta[max_ind3], max_p3-0.15*max_p3),\n",
    "                arrowprops=dict(facecolor='purple', shrink=2),\n",
    "                )\n",
    "    plt.xlabel(r'$\\theta$')\n",
    "    plt.ylabel(r'$P(θ|D)$')\n",
    "    plt.title('Posterior Probability of getting Heads')\n",
    "    \n",
    "    plt.show()\n",
    "\n",
    "    #Switch to display from print as it is easier on the eye\n",
    "    display(Math(r'\\text{The probability of Heads, } P(H)=\\theta,\\text{ using the Maximum Likelihood Estimate is } %s' % theta[maxL_ind]))\n",
    "    display(Math(r'\\text{The expected value of } P(H)=\\theta \\text{ using the Prior Probability is } 0.5'))\n",
    "    display(Math(r'\\text{The probability of Heads, }  P(H)=\\theta, \\text{ using the Bayesian Maximum A-Posteriori Estimate is } %s' % theta[max_ind3]))\n",
    "    return \n",
    "\n",
    "#set up of widgets with function plt_ML_BAY_EST\n",
    "#sigma=(min,max,step)\n",
    "interact(plt_ML_BAY_EST, userinput = 'HHT', sigma = (0.01,0.5,0.09))\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "#### By experimenting with userinput and $\\sigma$ values we see that our results show that:\n",
    "\n",
    "* When the prior distribution is more certain (narrow / small variance) than the ML distribution, we partly ignore our data in favour of trusting our prior knowledge and the Bayesian MAP extimate is closer to the prior expectation.\n",
    "\n",
    "* By contrast, when the prior distribution is more uncertain (wide/ large variance) than the ML distribution, we partly ignore our prior information in favour of trusting the data and the Bayesian MAP extimate is closer to the ML estimate."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
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